Kummer Theory

Kummer theory is an important branch of algebraic number theory that deals with the study of abelian extensions of number fields. It was introduced by Ernst Eduard Kummer in the 19th century as a tool for understanding the solvability of certain equations and has since become an essential tool for studying class field theory and the arithmetic of elliptic curves.

Background

In algebraic number theory, a number field is a finite extension of the rational numbers. The Galois group of a number field is the group of automorphisms that preserve the field structure and fix the rational numbers. The study of Galois groups is a fundamental aspect of algebraic number theory, as it allows us to understand the structure of number fields and their extensions.

Kummer theory deals specifically with the Galois group of abelian extensions of number fields. An abelian extension is a Galois extension whose Galois group is abelian. This means that the elements of the group can be ordered in such a way that the product of any two elements is equal to the product of their inverses.

Kummer Extensions

Kummer theory focuses on a particular type of abelian extension called a Kummer extension. A Kummer extension is a Galois extension obtained by adjoining the roots of a polynomial of the form $x^n-a$, where $a$ is an element of the base field and $n$ is a positive integer that is relatively prime to the characteristic of the base field.

The Galois group of a Kummer extension is isomorphic to a subgroup of the group $(\mathbb{Z}/n\mathbb{Z})^*$, where $(\mathbb{Z}/n\mathbb{Z})^*$ is the group of units modulo $n$. This group is abelian, so any Kummer extension is an abelian extension.

Kummer Theory

Kummer theory provides a powerful tool for understanding the structure of abelian extensions of number fields, particularly Kummer extensions. One of the key results of Kummer theory is that the solvability of certain polynomial equations depends on the existence of Kummer extensions with certain properties.

For example, consider the equation $x^3-2=0$. This equation has no rational roots, so it is not solvable in the rational numbers. However, if we adjoin a cube root of 2 to the rational numbers, we obtain a Kummer extension of degree 3. The Galois group of this extension is isomorphic to $(\mathbb{Z}/3\mathbb{Z})^*$, which is a cyclic group of order 2. It turns out that the equation $x^3-2=0$ is solvable in this extension, and in fact, every Kummer extension of degree 3 contains a root of this equation.

Kummer theory also has important applications in the study of class field theory and elliptic curves. One of the key results of class field theory is the existence of abelian extensions with certain properties, and Kummer theory provides a useful tool for constructing these extensions. Similarly, the arithmetic of elliptic curves is closely related to the study of abelian extensions, and Kummer theory plays a fundamental role in understanding the arithmetic of elliptic curves.

Conclusion

Kummer theory is an important branch of algebraic number theory that provides a powerful tool for understanding the structure of abelian extensions of number fields. Its applications are widespread, ranging from the study of polynomial equations to the arithmetic of elliptic curves. As such, it remains an active area of research and a fundamental tool for algebraic number theorists.

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